On subadditivity of Kodaira dimension in positive characteristic over a general type base
Author:
Zsolt Patakfalvi
Journal:
J. Algebraic Geom. 27 (2018), 21-53
DOI:
https://doi.org/10.1090/jag/688
Published electronically:
January 13, 2017
MathSciNet review:
3722689
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We show that for a surjective, separable morphism $f$ of smooth projective varieties over a field of positive characteristic such that $f_* \mathcal {O}_X \cong \mathcal {O}_Y$ subadditivity of Kodaira dimension holds, provided the base is of general type and the Hasse-Witt matrix of the geometric general fiber is not nilpotent.
References
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- Manuel Blickle, The $D$-module structure of $R[F]$-modules, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1647–1668. MR 1946409, DOI https://doi.org/10.1090/S0002-9947-02-03197-5
- Manuel Blickle and Gebhard Böckle, Cartier modules: finiteness results, J. Reine Angew. Math. 661 (2011), 85–123. MR 2863904, DOI https://doi.org/10.1515/CRELLE.2011.087
- Manuel Blickle and Karl Schwede, $p^{-1}$-linear maps in algebra and geometry, Commutative algebra, Springer, New York, 2013, pp. 123–205. MR 3051373, DOI https://doi.org/10.1007/978-1-4614-5292-8_5
- Jungkai Alfred Chen and Christopher D. Hacon, Kodaira dimension of irregular varieties, Invent. Math. 186 (2011), no. 3, 481–500. MR 2854084, DOI https://doi.org/10.1007/s00222-011-0323-x
- Yifei Chen and Lei Zhang, The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics, Math. Res. Lett. 22 (2015), no. 3, 675–696. MR 3350099, DOI https://doi.org/10.4310/MRL.2015.v22.n3.a3
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math. 12 (2001), no. 6, 689–741. MR 1875649, DOI https://doi.org/10.1142/S0129167X01000861
- Osamu Fujino, Algebraic fiber spaces whose general fibers are of maximal Albanese dimension, Nagoya Math. J. 172 (2003), 111–127. MR 2019522, DOI https://doi.org/10.1017/S0027763000008667
- Osamu Fujino, On maximal Albanese dimensional varieties, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 8, 92–95. MR 3127923, DOI https://doi.org/10.3792/pjaa.89.92
- Ofer Gabber, Notes on some $t$-structures, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 711–734. MR 2099084
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
- Christopher D. Hacon and Chenyang Xu, On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), no. 3, 711–744. MR 3327534, DOI https://doi.org/10.1090/S0894-0347-2014-00809-2
- Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Robin Hartshorne, Generalized divisors on Gorenstein schemes, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 287–339. MR 1291023, DOI https://doi.org/10.1007/BF00960866
- Shigeru Iitaka, Genus and classification of algebraic varieties. I, Sūgaku 24 (1972), no. 1, 14–27 (Japanese). MR 569689
- Yujiro Kawamata, The Kodaira dimension of certain fiber spaces, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 10, 406–408. MR 559042
- Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
- Yujiro Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982), no. 1, 57–71. MR 652646, DOI https://doi.org/10.1007/BF01404756
- Yujiro Kawamata, Hodge theory and Kodaira dimension, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 317–327. MR 715655, DOI https://doi.org/10.2969/aspm/00110317
- Yujiro Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. MR 814013, DOI https://doi.org/10.1515/crll.1985.363.1
- Dennis S. Keeler, Fujita’s conjecture and Frobenius amplitude, Amer. J. Math. 130 (2008), no. 5, 1327–1336. MR 2450210, DOI https://doi.org/10.1353/ajm.0.0015
- Neal Koblitz, $p$-adic variation of the zeta-function over families of varieties defined over finite fields, Compositio Math. 31 (1975), no. 2, 119–218. MR 414557
- János Kollár, Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 361–398. MR 946244, DOI https://doi.org/10.2969/aspm/01010361
- Ching-Jui Lai, Varieties fibered by good minimal models, Math. Ann. 350 (2011), no. 3, 533–547. MR 2805635, DOI https://doi.org/10.1007/s00208-010-0574-7
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- Zhao Hua Luo, Kodaira dimension of algebraic function fields, Amer. J. Math. 109 (1987), no. 4, 669–693. MR 900035, DOI https://doi.org/10.2307/2374609
- Gennady Lyubeznik, $F$-modules: applications to local cohomology and $D$-modules in characteristic $p>0$, J. Reine Angew. Math. 491 (1997), 65–130. MR 1476089, DOI https://doi.org/10.1515/crll.1997.491.65
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- Lance Edward Miller and Karl Schwede, Semi-log canonical vs $F$-pure singularities, J. Algebra 349 (2012), 150–164. MR 2853631, DOI https://doi.org/10.1016/j.jalgebra.2011.08.035
- L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur $\mathbb {P}^ 1$ (II. Exemples), Astérisque (1981), no. 86, 125–140.
- Shigefumi Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 269–331. MR 927961
- Mircea Mustaţă and Vasudevan Srinivas, Ordinary varieties and the comparison between multiplier ideals and test ideals, Nagoya Math. J. 204 (2011), 125–157. MR 2863367, DOI https://doi.org/10.1215/00277630-1431849
- Zsolt Patakfalvi, Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 5, 991–1025 (English, with English and French summaries). MR 3294622, DOI https://doi.org/10.24033/asens.2232
- Zsolt Patakfalvi and Karl Schwede, Depth of $F$-singularities and base change of relative canonical sheaves, J. Inst. Math. Jussieu 13 (2014), no. 1, 43–63. MR 3134015, DOI https://doi.org/10.1017/S1474748013000066
- Zs. Patakfalvi, K. Schwede, and W. Zhang, $F$-singularities in families, http://arxiv.org/abs/1305.1646 (2013).
- Karl Schwede, A canonical linear system associated to adjoint divisors in characteristic $p>0$, J. Reine Angew. Math. 696 (2014), 69–87. MR 3276163, DOI https://doi.org/10.1515/crelle-2012-0087
- Kenji Ueno, Classification of algebraic varieties. I, Compositio Math. 27 (1973), 277–342. MR 360582
- Kenji Ueno, On algebraic fibre spaces of abelian varieties, Math. Ann. 237 (1978), no. 1, 1–22. MR 506652, DOI https://doi.org/10.1007/BF01351555
- Eckart Viehweg, Canonical divisors and the additivity of the Kodaira dimension for morphisms of relative dimension one, Compositio Math. 35 (1977), no. 2, 197–223. MR 569690
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329–353. MR 715656, DOI https://doi.org/10.2969/aspm/00110329
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension. II. The local Torelli map, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 567–589. MR 728619
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
References
- Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
- Lucian Bădescu, Algebraic surfaces, Universitext, Springer-Verlag, New York, 2001. Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author. MR 1805816
- Caucher Birkar, The Iitaka conjecture $C_{n,m}$ in dimension six, Compos. Math. 145 (2009), no. 6, 1442–1446. MR 2575089, DOI https://doi.org/10.1112/S0010437X09004187
- Manuel Blickle, The $D$-module structure of $R[F]$-modules, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1647–1668. MR 1946409, DOI https://doi.org/10.1090/S0002-9947-02-03197-5
- Manuel Blickle and Gebhard Böckle, Cartier modules: finiteness results, J. Reine Angew. Math. 661 (2011), 85–123. MR 2863904, DOI https://doi.org/10.1515/CRELLE.2011.087
- Manuel Blickle and Karl Schwede, $p^{-1}$-linear maps in algebra and geometry, Commutative algebra, Springer, New York, 2013, pp. 123–205. MR 3051373, DOI https://doi.org/10.1007/978-1-4614-5292-8_5
- Jungkai Alfred Chen and Christopher D. Hacon, Kodaira dimension of irregular varieties, Invent. Math. 186 (2011), no. 3, 481–500. MR 2854084, DOI https://doi.org/10.1007/s00222-011-0323-x
- Yifei Chen and Lei Zhang, The subadditivity of the Kodaira dimension for fibrations of relative dimension one in positive characteristics, Math. Res. Lett. 22 (2015), no. 3, 675–696. MR 3350099, DOI https://doi.org/10.4310/MRL.2015.v22.n3.a3
- Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math. 12 (2001), no. 6, 689–741. MR 1875649, DOI https://doi.org/10.1142/S0129167X01000861
- Osamu Fujino, Algebraic fiber spaces whose general fibers are of maximal Albanese dimension, Nagoya Math. J. 172 (2003), 111–127. MR 2019522
- Osamu Fujino, On maximal Albanese dimensional varieties, Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 8, 92–95. MR 3127923, DOI https://doi.org/10.3792/pjaa.89.92
- Ofer Gabber, Notes on some $t$-structures, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 711–734. MR 2099084
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 0199181
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 0217086
- Christopher D. Hacon and Chenyang Xu, On the three dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), no. 3, 711–744. MR 3327534, DOI https://doi.org/10.1090/S0894-0347-2014-00809-2
- Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. MR 0222093
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
- Robin Hartshorne, Generalized divisors on Gorenstein schemes, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 287–339. MR 1291023, DOI https://doi.org/10.1007/BF00960866
- Shigeru Iitaka, Genus and classification of algebraic varieties. I, Sûgaku 24 (1972), no. 1, 14–27 (Japanese). MR 0569689
- Yujiro Kawamata, The Kodaira dimension of certain fiber spaces, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 10, 406–408. MR 559042
- Yujiro Kawamata, Characterization of abelian varieties, Compositio Math. 43 (1981), no. 2, 253–276. MR 622451
- Yujiro Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982), no. 1, 57–71. MR 652646, DOI https://doi.org/10.1007/BF01404756
- Yujiro Kawamata, Hodge theory and Kodaira dimension, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 317–327. MR 715655
- Yujiro Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math. 363 (1985), 1–46. MR 814013, DOI https://doi.org/10.1515/crll.1985.363.1
- Dennis S. Keeler, Fujita’s conjecture and Frobenius amplitude, Amer. J. Math. 130 (2008), no. 5, 1327–1336. MR 2450210, DOI https://doi.org/10.1353/ajm.0.0015
- Neal Koblitz, $p$-adic variation of the zeta-function over families of varieties defined over finite fields, Compositio Math. 31 (1975), no. 2, 119–218. MR 0414557
- János Kollár, Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 361–398. MR 946244
- Ching-Jui Lai, Varieties fibered by good minimal models, Math. Ann. 350 (2011), no. 3, 533–547. MR 2805635, DOI https://doi.org/10.1007/s00208-010-0574-7
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. MR 2095471
- Zhao Hua Luo, Kodaira dimension of algebraic function fields, Amer. J. Math. 109 (1987), no. 4, 669–693. MR 900035, DOI https://doi.org/10.2307/2374609
- Gennady Lyubeznik, $F$-modules: applications to local cohomology and $D$-modules in characteristic $p>0$, J. Reine Angew. Math. 491 (1997), 65–130. MR 1476089, DOI https://doi.org/10.1515/crll.1997.491.65
- Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
- Lance Edward Miller and Karl Schwede, Semi-log canonical vs $F$-pure singularities, J. Algebra 349 (2012), 150–164. MR 2853631, DOI https://doi.org/10.1016/j.jalgebra.2011.08.035
- L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur $\mathbb {P}^ 1$ (II. Exemples), Astérisque (1981), no. 86, 125–140.
- Shigefumi Mori, Classification of higher-dimensional varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 269–331. MR 927961
- Mircea Mustaţă and Vasudevan Srinivas, Ordinary varieties and the comparison between multiplier ideals and test ideals, Nagoya Math. J. 204 (2011), 125–157. MR 2863367
- Zsolt Patakfalvi, Semi-positivity in positive characteristics, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 5, 991–1025 (English, with English and French summaries). MR 3294622
- Zsolt Patakfalvi and Karl Schwede, Depth of $F$-singularities and base change of relative canonical sheaves, J. Inst. Math. Jussieu 13 (2014), no. 1, 43–63. MR 3134015, DOI https://doi.org/10.1017/S1474748013000066
- Zs. Patakfalvi, K. Schwede, and W. Zhang, $F$-singularities in families, http://arxiv.org/abs/1305.1646 (2013).
- Karl Schwede, A canonical linear system associated to adjoint divisors in characteristic $p>0$, J. Reine Angew. Math. 696 (2014), 69–87. MR 3276163, DOI https://doi.org/10.1515/crelle-2012-0087
- Kenji Ueno, Classification of algebraic varieties. I, Compositio Math. 27 (1973), 277–342. MR 0360582
- Kenji Ueno, On algebraic fibre spaces of abelian varieties, Math. Ann. 237 (1978), no. 1, 1–22. MR 506652, DOI https://doi.org/10.1007/BF01351555
- Eckart Viehweg, Canonical divisors and the additivity of the Kodaira dimension for morphisms of relative dimension one, Compositio Math. 35 (1977), no. 2, 197–223. MR 0569690
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 329–353. MR 715656
- Eckart Viehweg, Weak positivity and the additivity of the Kodaira dimension. II. The local Torelli map, Classification of algebraic and analytic manifolds (Katata, 1982) Progr. Math., vol. 39, Birkhäuser Boston, Boston, MA, 1983, pp. 567–589. MR 728619
- Eckart Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 30, Springer-Verlag, Berlin, 1995. MR 1368632
Additional Information
Zsolt Patakfalvi
Affiliation:
Department of Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, New Jersey 08544-1000
Address at time of publication:
EPFL SB MATHGEOM CAG, MA B3 444 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland
Email:
zsolt.patakfalvi@epfl.ch
Received by editor(s):
May 1, 2015
Received by editor(s) in revised form:
February 12, 2016, and April 9, 2016
Published electronically:
January 13, 2017
Article copyright:
© Copyright 2017
University Press, Inc.